About the computer images on the poster (and in the logo)

Lorenz Manifolds

The computer generated images on the Equadiff  2003 poster show the two-dimensional stable manifold W^s(0) of the origin of the Lorenz system (for the classic parameter values). They were computed by Bernd Krauskopf and Hinke Osinga with the method in [1].

Starting with a small circle in the linear stable eigenspace around the origin, the algorithm computes a set of  approximate level sets of the geodesic distance along W^s(0) from the origin. These level sets are topological circles. The manifold W^s(0) itself is constructed as a set of ribbons between consecutive computed geodesic level sets. Points are added or removed adaptively according to prespecified accuracy parameters, so that a good and uniform mesh quality is achieved. In total 72 level sets were computed, and the last one is approximately at geodesic distance 151.75 from the origin.

The color images show the entire manifold from two different viewpoints. Only every second ribbon is shown, providing a see-through effect. The rainbow color scheme indicates geodesic distance from the origin, from blue (close) via green to red (far).

The blue tinted background image shows only the part of W^s(0) inside a sphere of radius 60 around the point (0,0,27). This allows one to see how the manifold spirals into the famous chaotic attractor of the Lorenz system.

More on the visualization of W^s(0) can be found in [2].
 

[1] B. Krauskopf and H.M. Osinga, Two-dimensional global manifolds of vector fields, CHAOS 9 (3) (1999) 768-774; see also the multimedia supplement posted EPAPS.

[2] H.M. Osinga and B. Krauskopf, Visualizing chaos in the Lorenz system, Computer and Graphics 26 (5) (2002) 815-823.